For what value of a, the divisions of two polynomials (ax³ + 3x² - 3) and (2x³ - 5x + a) by (x - 4) give the same remainder.
Let us calculate and write it.
Answers
The value of a is 1 respectively.
Solution:-
In the first case,
f(x)= ax³ + 3x² - 3 when divided by the linear polynomial (x - 4)
At first, we will find out the zero of the polynomial (x - 4):-
=> x - 4 = 0
=> x = 4
From the remainder theorem,
The required remainder = f(4)
Putting the value of x,
= ax³ + 3x² - 3
= a.(4)³ + 3.(4)² - 3
= 64a + 48 - 3
= 64a + 45 ........(i)
In the second case,
g(x) = 2x³ - 5x + a when divided by the linear polynomial (x - 4)
Zero of the polynomial (x - 4) = 4
Putting the value of x
= 2x³ - 5x + a
= 2.(4)³ - 5.4 + a
= 2.64 - 20 + a
= 128 - 20 + a
= a + 108 ........(ii)
By comparing eqn.(i) & (ii), we get:-
=> 64a + 45 = a + 108
=> 64a - a = 108 - 45
=> 63a = 63
=> a = 63/63
=> a = 1
Hence, the value of a is 1.
Verification:-
Putting the value of a and x in the polynomial f(x), we get:-
= ax³ + 3x² - 3
= 1.(4)³ + 3.(4)² - 3
= 64 + 48 - 3
= 109
Putting the value of a and x in the polynomial g(x), we get:-
= 2x³ - 5x + a
= 2.(4)³ - 5.4 + 1
= 128 - 20 + 1
= 109
Therefore, the remainders arethe same.
Hence, proved.
Answer:
Value of a = 1
Step-by-step explanation:
Given Problem:
For what value of a, the divisions of two polynomials (ax³ + 3x² - 3) and (2x³ - 5x + a) by (x - 4) give the same remainder.
Let us calculate and write it.
Solution:
To Find:
Value of a.
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Method:
Given that polynomials are:
It is also given that these two polynomials leave the same remainder when divided by
So,